1 Answer
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No, this isn't possible. What you have is invariant under rotations, but the lengths of the green sides aren't. You need something that fixes the orientation, ideally the angle that the chord makes with one of the axes or something similar.

And say I have the coordinates of the left most black dot? Then it should be possible right?
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JonasDec 15 '11 at 12:13

Yes. In that case, it would be easiest to translate your coordinate system such that the origin lies at the centre of the circle (i.e. subtract the radius from all coordinates); then you could calculate the angle at which that black dot lies from the centre as the arctangent of the ratio of its coordinates, and then add the arctangent of the ratio of $dB$ to $h$ to get the angle $\alpha$ at which the red dot lies from the centre; then its coordinates (in the translated coordinate system) are $(h\cos\alpha,h\sin\alpha)$.
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jorikiDec 15 '11 at 12:23

You'll need to take some care to get the signs right. If your computing environment has an atan2 function, you should use that; otherwise you might have to make some case distinctions.
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jorikiDec 15 '11 at 12:27